State-space model realization for non-commensurate fractional-order systems based on Gleason’s problem

Abstract

This paper presents a novel state-space model (SSM) realization approach for non-commensurate fractional-order (NCFO) systems. A new notion of the resolvent invariant backward shift space of Gleason’s problem associated with the NCFO rational transfer function matrix is introduced firstly, based on which a necessary and sufficient condition for the existence of the NCFO-SSM realization is given. Then, a sufficient condition for the construction of the resolvent invariant backward shift space of NCFO Gleason’s problem is presented, and the corresponding realization algorithms are developed. In order to obtain a lower-dimensional NCFO-SSM, further improvement based on a polynomial description is studied. Finally, symbolic and numerical examples are provided to illustrate the details as well as the effectiveness of the proposed NCFO-SSM realization approach.